The dynamic convex hull problem was recently solved in
O(lg n) time per operation, and this result is best
possible in models of computation with bounded branching (e.g., algebraic
computation trees). From a data structures point of view, however, such
models are considered unrealistic because they hide intrinsic notions of
information in the input.

In the standard word-RAM and cell-probe models of computation, we prove that
the optimal query time for dynamic convex hulls is, in fact,
Θ(lg n / lg lg n), for
polylogarithmic update time (and word size). Our lower bound is based on a
reduction from the marked-ancestor problem, and is one of the first data
structural lower bounds for a nonorthogonal geometric problem. Our upper
bounds follow a recent trend of attacking nonorthogonal geometric problems
from an information-theoretic perspective that has proved central to advanced
data structures. Interestingly, our upper bounds are the first to
successfully apply this perspective to dynamic geometric data structures, and
require substantially different ideas from previous work.